Simple proofs of certain results on generalized Fekete-Szeg\H{o} functional in the class $\boldsymbol{\mathcal{S}}$

TL;DR

This paper provides simplified, sharp proofs for bounds on the generalized Fekete-Szeg ext{"o} functional for univalent and convex functions, improving understanding of coefficient inequalities in these classes.

Contribution

It introduces easier proof techniques for key inequalities of the generalized Fekete-Szeg ext{"o} functional, applicable to both $ ext{S}$ and $ ext{K}$ classes, with sharp bounds.

Findings

Established sharp bounds for the functional in $ ext{S}$ and $ ext{K}$ classes.

Provided simplified proofs leveraging known univalent function results.

Extended results to both the entire class and convex subclass.

Abstract

In this paper we give simple proofs for the main results concerning generalized Fekete-Szeg\H{o} functional of type $\left|a_{3}(f)-\lambda a_{2}(f)^{2}\right|-\mu|a_{2}(f)|$, where $\lambda\in\mathbb{C}$, $\mu>0$ and $a_{n}(f)$ is $n$-th coefficient of the power series expansion of $f\in\mathcal{S}$. In addition, we studied this functional separately for the class $\mathcal{K}$ of convex functions and we emphasize that all the results of the paper are sharp (i.e. the best possible). The advantages of the present study are that the techniques used in the proofs are more easier and use known results regarding the univalent functions, and those that it give the best possible results not only for the entire class of univalent normalized functions $\mathcal{S}$ but also for its subclass of convex functions $\mathcal{K}$.